1. Number Representations

2. Numbers with Fractions
So, done with negative numbers. Done with signed and unsigned integers.
How about numbers with fractions?
How to represent, say, 5.75ten in binary forms?

3. Numbers with Fractions
In general, to represent a real number in binary, you first find the binary representation of the integer part, then find the binary representation of the fraction part, then put a dot in between.

4. Numbers with fractions
The integer part is 5ten which is 101two. How did you get it?

5. Numbers with Fractions
The fraction is Note that it is = , so
5.75ten = two

6. How to get the fraction
In general, what you do is kind of the reverse of getting the binary representation for the integer: divide the fraction first by 0.5 (2-1), take the quotient as the first bit of the binary fraction, then divide the remainder by 0.25 (2-2), take the quotient as the second bit of the binary fraction, then divide the remainder by (2-3),

7. How to get the fraction Take 0.1 as an example.
0.1/0.5=0* –> bit 1 is 0.
0.1/0.25 = 0* –> bit 2 is 0.
0.1/0.125 = 0* –> bit 3 is 0.
0.1/ = 1* –> bit 4 is 1.
0.0375/ = 1* –> bit 5 is 1.
And so on, until the you have used all the bits that hardware permits

8. Floating Point Numbers
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Floating Point Numbers
Recall scientific notation for decimal numbers
A number is represented by a significand (or mantissa) and an integer exponent F * 10E
Where F is the significand, and E the exponent
Example:
* 102
Normalized if F is a single digit number
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9. Floating Points in Binary
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Floating Points in Binary
Normalized binary scientific notation
For a fixed number of bits, we need to decide
How many bits for the significand (or fraction)
How many bits for the exponent
There is a trade-off between precision and range
More bits for significand increases precision while more bits for exponent increases the range
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10. IEEE 754 Floating Point Standard
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IEEE 754 Floating Point Standard
Single precision
Represented by 32 bits
Since the leading 1 bit in the significand in normalized binary numbers is always 1, it is not represented explicitly
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11. 11/19/2018
Exponent
If we represent exponents using two’s complement, then it would not be intuitive as small numbers appear to be larger 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
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12. 11/19/2018
Biased Notation
The most negative exponent will be represented as 00…00 and the most positive as 111…11
That is, we need to subtract the bias from the corresponding unassigned value
The value of an IEEE 754 single precision is 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 s
exponent
fraction
1 bit
8 bits
23 bits
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13. Example 101.11two= 22 + 20 + 2-1 + 2-2 = 5.75ten
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Example
101.11two= = 5.75ten
The normalized binary number will be 22 = 2( )
So the exponent is 129ten =
As a hexadecimal number, the representation is
0x40B80000 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
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14. IEEE 754 Double Precision It uses 64 bits (two 32-bit words)
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IEEE 754 Double Precision
It uses 64 bits (two 32-bit words)
1 bit for the sign
11 bits for the exponent
52 bits for the fraction
1023 as the bias 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 s
Exponent
fraction
1 bit
11 bits
20 bits
Fraction (continued)
32 bits
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15. Example (Double Precision)
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Example (Double Precision)
101.11two= = 5.75
The normalized binary number will be 22 = 2( )
So the exponent is 1025ten = two
As a hexadecimal number, the representation is 0x 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
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16. floating-point number
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Special Cases
Single precision
Double precision
Object represented
Exponent
Fraction
nonzero
denormalized number
1-254
anything
1-2046
floating-point number
255
2047
 infinity
NaN (Not a number)
Denormalized
If the exponent is all 0s, but the fraction is non-zero (else it would be interpreted as zero), then the value is a denormalized number, which does not have an assumed leading 1 before the binary point. Thus, this represents a number (-1)s × 0.f × 2-126, where s is the sign bit and f is the fraction. For double precision, denormalized numbers are of the form (-1)s × 0.f × From this you can interpret zero as a special type of denormalized number.
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17. Floating Point Numbers
How many different numbers can the single precision format represent? What is the largest number it can represent?

18. Ranges for IEEE 754 Single Precision
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Ranges for IEEE 754 Single Precision
Largest positive number
Smallest positive number (floating point) 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
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19. Ranges for IEEE 754 Single Precision
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Ranges for IEEE 754 Single Precision
Largest positive number
Smallest positive number (floating point)
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20. Ranges for IEEE 754 Double Precision
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Ranges for IEEE 754 Double Precision
Largest positive number
Smallest positive number (Floating-point number)
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21. Comments on Overflow and Underflow
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Comments on Overflow and Underflow
Overflow (and underflow also for floating numbers) happens when a number is outside the range of a particular representation
For example, by using 8-bit two’s complement representation, we can only represent a number between and 127
If a number is smaller than -128, it will cause overflow
If a number is larger than 127, it will cause overflow also
Note that arithmetic operations can result in overflow
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